Ebook: Mathematical Methods in Tomography: Proceedings of a Conference held in Oberwolfach, Germany, 5–11 June, 1990

The conference was devoted to the discussion of present and future techniques in medical imaging, including 3D x-ray CT, ultrasound and diffraction tomography, and biomagnetic ima- ging. The mathematical models, their theoretical aspects and the development of algorithms were treated. The proceedings contains surveys on reconstruction in inverse obstacle scat- tering, inversion in 3D, and constrained least squares pro- blems.Research papers include besides the mentioned imaging techniques presentations on image reconstruction in Hilbert spaces, singular value decompositions, 3D cone beam recon- struction, diffuse tomography, regularization of ill-posed problems, evaluation reconstruction algorithms and applica- tions in non-medical fields. Contents: Theoretical Aspects: J.Boman: Helgason' s support theorem for Radon transforms-a newproof and a generalization -P.Maass: Singular value de- compositions for Radon transforms- W.R.Madych: Image recon- struction in Hilbert space -R.G.Mukhometov: A problem of in- tegral geometry for a family of rays with multiple reflec- tions -V.P.Palamodov: Inversion formulas for the three-di- mensional ray transform - Medical Imaging Techniques: V.Friedrich: Backscattered Photons - are they useful for a surface - near tomography - P.Grangeat: Mathematical frame- work of cone beam 3D reconstruction via the first derivative of the Radon transform -P.Grassin,B.Duchene,W.Tabbara: Dif- fraction tomography: some applications and extension to 3D ultrasound imaging -F.A.Gr}nbaum: Diffuse tomography: a re- fined model -R.Kress,A.Zinn: Three dimensional reconstruc- tions in inverse obstacle scattering -A.K.Louis: Mathemati- cal questions of a biomagnetic imaging problem - Inverse Problems and Optimization: Y.Censor: On variable block algebraic reconstruction techniques -P.P.Eggermont: On Volterra-Lotka differential equations and multiplicative algorithms for monotone complementary problems

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The conference was devoted to the discussion of present and future techniques in medical imaging, including 3D x-ray CT, ultrasound and diffraction tomography, and biomagnetic ima- ging. The mathematical models, their theoretical aspects and the development of algorithms were treated. The proceedings contains surveys on reconstruction in inverse obstacle scat- tering, inversion in 3D, and constrained least squares pro- blems.Research papers include besides the mentioned imaging techniques presentations on image reconstruction in Hilbert spaces, singular value decompositions, 3D cone beam recon- struction, diffuse tomography, regularization of ill-posed problems, evaluation reconstruction algorithms and applica- tions in non-medical fields. Contents: Theoretical Aspects: J.Boman: Helgason' s support theorem for Radon transforms-a newproof and a generalization -P.Maass: Singular value de- compositions for Radon transforms- W.R.Madych: Image recon- struction in Hilbert space -R.G.Mukhometov: A problem of in- tegral geometry for a family of rays with multiple reflec- tions -V.P.Palamodov: Inversion formulas for the three-di- mensional ray transform - Medical Imaging Techniques: V.Friedrich: Backscattered Photons - are they useful for a surface - near tomography - P.Grangeat: Mathematical frame- work of cone beam 3D reconstruction via the first derivative of the Radon transform -P.Grassin,B.Duchene,W.Tabbara: Dif- fraction tomography: some applications and extension to 3D ultrasound imaging -F.A.Gr}nbaum: Diffuse tomography: a re- fined model -R.Kress,A.Zinn: Three dimensional reconstruc- tions in inverse obstacle scattering -A.K.Louis: Mathemati- cal questions of a biomagnetic imaging problem - Inverse Problems and Optimization: Y.Censor: On variable block algebraic reconstruction techniques -P.P.Eggermont: On Volterra-Lotka differential equations and multiplicative algorithms for monotone complementary problems

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The conference was devoted to the discussion of present and future techniques in medical imaging, including 3D x-ray CT, ultrasound and diffraction tomography, and biomagnetic ima- ging. The mathematical models, their theoretical aspects and the development of algorithms were treated. The proceedings contains surveys on reconstruction in inverse obstacle scat- tering, inversion in 3D, and constrained least squares pro- blems.Research papers include besides the mentioned imaging techniques presentations on image reconstruction in Hilbert spaces, singular value decompositions, 3D cone beam recon- struction, diffuse tomography, regularization of ill-posed problems, evaluation reconstruction algorithms and applica- tions in non-medical fields. Contents: Theoretical Aspects: J.Boman: Helgason' s support theorem for Radon transforms-a newproof and a generalization -P.Maass: Singular value de- compositions for Radon transforms- W.R.Madych: Image recon- struction in Hilbert space -R.G.Mukhometov: A problem of in- tegral geometry for a family of rays with multiple reflec- tions -V.P.Palamodov: Inversion formulas for the three-di- mensional ray transform - Medical Imaging Techniques: V.Friedrich: Backscattered Photons - are they useful for a surface - near tomography - P.Grangeat: Mathematical frame- work of cone beam 3D reconstruction via the first derivative of the Radon transform -P.Grassin,B.Duchene,W.Tabbara: Dif- fraction tomography: some applications and extension to 3D ultrasound imaging -F.A.Gr}nbaum: Diffuse tomography: a re- fined model -R.Kress,A.Zinn: Three dimensional reconstruc- tions in inverse obstacle scattering -A.K.Louis: Mathemati- cal questions of a biomagnetic imaging problem - Inverse Problems and Optimization: Y.Censor: On variable block algebraic reconstruction techniques -P.P.Eggermont: On Volterra-Lotka differential equations and multiplicative algorithms for monotone complementary problems
Content:
Front Matter....Pages I-X
Helgason's support theorem for Radon transforms — A new proof and a generalization....Pages 1-5
Singular value decompositions for Radon transforms....Pages 6-14
Image reconstruction in Hilbert space....Pages 15-45
A problem of integral geometry for a family of rays with multiple reflections....Pages 46-52
Inversion formulas for the three-dimensional ray transform....Pages 53-62
Backscattered photons — Are they useful for a surface-near tomography?....Pages 63-65
Mathematical framework of cone beam 3D reconstruction via the first derivative of the radon transform....Pages 66-97
Diffraction tomography some applications and extension to 3-D ultrasound imaging....Pages 98-105
Diffuse tomography: A refined model....Pages 106-111
Three dimensional reconstructions in inverse obstacle scattering....Pages 112-125
Mathematical questions of a biomagnetic imaging problem....Pages 126-132
On variable block algebraic reconstruction techniques....Pages 133-140
On Volterra-Lotka differential equations and multiplicative algorithms for monotone complementarity problems....Pages 141-152
Constrained regularized least squares problems....Pages 153-166
Multiplicative iterative methods in computed tomography....Pages 167-186
Remark on the informative content of few measurements....Pages 187-193
Theorems for the number of zeros of the projection radial modulators of the 2D exponential radon transform....Pages 194-214
Evaluation of reconstruction algorithms....Pages 215-228
Radon transform and analog coding....Pages 229-241
Determination of the specific density of an aerosol through tomography....Pages 242-260
Computed tomography and rockets....Pages 261-268

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The conference was devoted to the discussion of present and future techniques in medical imaging, including 3D x-ray CT, ultrasound and diffraction tomography, and biomagnetic ima- ging. The mathematical models, their theoretical aspects and the development of algorithms were treated. The proceedings contains surveys on reconstruction in inverse obstacle scat- tering, inversion in 3D, and constrained least squares pro- blems.Research papers include besides the mentioned imaging techniques presentations on image reconstruction in Hilbert spaces, singular value decompositions, 3D cone beam recon- struction, diffuse tomography, regularization of ill-posed problems, evaluation reconstruction algorithms and applica- tions in non-medical fields. Contents: Theoretical Aspects: J.Boman: Helgason' s support theorem for Radon transforms-a newproof and a generalization -P.Maass: Singular value de- compositions for Radon transforms- W.R.Madych: Image recon- struction in Hilbert space -R.G.Mukhometov: A problem of in- tegral geometry for a family of rays with multiple reflec- tions -V.P.Palamodov: Inversion formulas for the three-di- mensional ray transform - Medical Imaging Techniques: V.Friedrich: Backscattered Photons - are they useful for a surface - near tomography - P.Grangeat: Mathematical frame- work of cone beam 3D reconstruction via the first derivative of the Radon transform -P.Grassin,B.Duchene,W.Tabbara: Dif- fraction tomography: some applications and extension to 3D ultrasound imaging -F.A.Gr}nbaum: Diffuse tomography: a re- fined model -R.Kress,A.Zinn: Three dimensional reconstruc- tions in inverse obstacle scattering -A.K.Louis: Mathemati- cal questions of a biomagnetic imaging problem - Inverse Problems and Optimization: Y.Censor: On variable block algebraic reconstruction techniques -P.P.Eggermont: On Volterra-Lotka differential equations and multiplicative algorithms for monotone complementary problems
Content:
Front Matter....Pages I-X
Helgason's support theorem for Radon transforms — A new proof and a generalization....Pages 1-5
Singular value decompositions for Radon transforms....Pages 6-14
Image reconstruction in Hilbert space....Pages 15-45
A problem of integral geometry for a family of rays with multiple reflections....Pages 46-52
Inversion formulas for the three-dimensional ray transform....Pages 53-62
Backscattered photons — Are they useful for a surface-near tomography?....Pages 63-65
Mathematical framework of cone beam 3D reconstruction via the first derivative of the radon transform....Pages 66-97
Diffraction tomography some applications and extension to 3-D ultrasound imaging....Pages 98-105
Diffuse tomography: A refined model....Pages 106-111
Three dimensional reconstructions in inverse obstacle scattering....Pages 112-125
Mathematical questions of a biomagnetic imaging problem....Pages 126-132
On variable block algebraic reconstruction techniques....Pages 133-140
On Volterra-Lotka differential equations and multiplicative algorithms for monotone complementarity problems....Pages 141-152
Constrained regularized least squares problems....Pages 153-166
Multiplicative iterative methods in computed tomography....Pages 167-186
Remark on the informative content of few measurements....Pages 187-193
Theorems for the number of zeros of the projection radial modulators of the 2D exponential radon transform....Pages 194-214
Evaluation of reconstruction algorithms....Pages 215-228
Radon transform and analog coding....Pages 229-241
Determination of the specific density of an aerosol through tomography....Pages 242-260
Computed tomography and rockets....Pages 261-268
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